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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|87}}
{{ED intro}}


== Theory ==
== Theory ==
87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is distinctly [[consistent]] in the [[13-odd-limit]] [[tonality diamond]], and also the smallest edo with [[relative interval error]]s of no greater than 25% on all of the first 16 harmonics of the [[harmonic series]]. It is also a [[zeta peak integer edo]].
87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is [[distinctly consistent]] in the [[13-odd-limit]] [[tonality diamond]], and the smallest edo that is [[purely consistent]]{{idiosyncratic}} in the [[15-odd-limit]] (maintains [[relative interval error]]s of no greater than 25% on all of the first 16 [[harmonic]]s of the [[harmonic series]]). It is also a [[zeta peak integer edo]]. Since {{nowrap|87 {{=}} 3 × 29}}, 87edo shares the same perfect fifth with [[29edo]].  


87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]] and [[29/1|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  
87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.  


The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 46 -29 }} ([[29-comma]]) in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].  
It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 46 -29 }} ([[29-comma]]) in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].  


87edo is a particularly good tuning for [[rodan]], the 41 & 46 temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE generator]] and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
87edo is a particularly good tuning for [[rodan]], the {{nowrap|41 & 46}} temperament. The 8/7 generator of 17\87 is a remarkable 0.00061{{c}} sharper than the 13-limit [[CWE tuning|CWE generator]]. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.01479{{c}} sharp of the 13-limit CWE generator.


=== Prime harmonics ===
=== Prime harmonics ===
In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[Square superparticular|S116 through S137]] by patent val and corresponding to the gravity of the fact that 87edo is a circle of [[126/125]]'s, meaning ([[126/125]])<sup>87</sup> only very slightly exceeds the octave.
In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[S-expression|S116 through S137]] by patent val and corresponding to the gravity of the fact that 87edo is a circle of [[126/125]]'s, meaning ([[126/125]])<sup>87</sup> only very slightly exceeds the octave.
{{Harmonics in equal|87|columns=12}}
{{Harmonics in equal|87|columns=12}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
=== Subsets and supersets ===
87edo contains [[3edo]] and [[29edo]] as subset edos.
[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic. Multiples like [[696edo]] and [[783edo]] are also of theoretical interest.
One step of 87edo is ''extremely'' close to [[126/125]], with the associated comma, 87-126/125-comma ({{monzo|86 174 -261 87}}) being tempered in equal temperaments well over 100,000 and with less than 0.001{{c}} error, with the [[optimal patent val]] for the 7-limit temperament being an enormous [[425082edo]]. But more notably, 87edo itself does the mapping consistently, making it a true circle of 126/125's.
See also [[87th-octave temperaments]].


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4"
{| class="wikitable center-all right-2 left-3 left-4"
|-
|-
! rowspan="2" | &#35;
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Cents
! colspan="2" | Approximated ratios
! colspan="2" | Approximated ratios
! colspan="2" rowspan="2" | [[Ups and downs notation]]
! colspan="2" rowspan="2" | [[Ups and downs notation]]
|-
|-
! 13-Limit
! 13-limit
! 31-Limit No-7s Extension
! 31-limit extension
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
|
|
Line 35: Line 44:
|-
|-
| 1
| 1
| 13.793
| 13.8
| [[91/90]], [[100/99]], [[126/125]]
| [[91/90]], [[100/99]], [[126/125]]
|
|
Line 42: Line 51:
|-
|-
| 2
| 2
| 27.586
| 27.6
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
| ''[[49/48]]'', [[55/54]], [[64/63]], [[65/64]], [[81/80]]
|
|
Line 49: Line 58:
|-
|-
| 3
| 3
| 41.379
| 41.4
| [[40/39]], [[45/44]], [[50/49]]
| [[40/39]], [[45/44]], [[50/49]]
| [[39/38]]
| [[39/38]]
Line 56: Line 65:
|-
|-
| 4
| 4
| 55.172
| 55.2
| ''[[28/27]]'', [[33/32]], [[36/35]]
| ''[[28/27]]'', [[33/32]], [[36/35]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
| [[30/29]], [[31/30]], [[32/31]], [[34/33]]
Line 63: Line 72:
|-
|-
| 5
| 5
| 68.966
| 69.0
| [[25/24]], [[26/25]], [[27/26]]
| [[25/24]], [[26/25]], [[27/26]]
| [[24/23]]
| [[24/23]]
Line 70: Line 79:
|-
|-
| 6
| 6
| 82.759
| 82.8
| [[21/20]], [[22/21]]
| [[21/20]], [[22/21]]
| [[20/19]], [[23/22]]
| [[20/19]], [[23/22]]
Line 77: Line 86:
|-
|-
| 7
| 7
| 96.552
| 96.6
| [[35/33]]
| [[35/33]]
| [[18/17]], [[19/18]]
| [[18/17]], [[19/18]]
Line 84: Line 93:
|-
|-
| 8
| 8
| 110.345
| 110.3
| [[16/15]]
| [[16/15]]
| [[17/16]], [[31/29]], [[33/31]]
| [[17/16]], [[31/29]], [[33/31]]
Line 91: Line 100:
|-
|-
| 9
| 9
| 124.138
| 124.1
| [[14/13]], [[15/14]]
| [[14/13]], [[15/14]]
| [[29/27]]
| [[29/27]]
Line 98: Line 107:
|-
|-
| 10
| 10
| 137.931
| 137.9
| [[13/12]], [[27/25]]
| [[13/12]], [[27/25]]
| [[25/23]]
| [[25/23]]
Line 105: Line 114:
|-
|-
| 11
| 11
| 151.724
| 151.7
| [[12/11]], [[35/32]]
| [[12/11]], [[35/32]]
|
|
Line 112: Line 121:
|-
|-
| 12
| 12
| 165.517
| 165.5
| [[11/10]]
| [[11/10]]
| [[32/29]], [[34/31]]
| [[32/29]], [[34/31]]
Line 119: Line 128:
|-
|-
| 13
| 13
| 179.310
| 179.3
| [[10/9]]
| [[10/9]]
|
|
Line 126: Line 135:
|-
|-
| 14
| 14
| 193.103
| 193.1
| [[28/25]]
| [[28/25]]
| [[19/17]], [[29/26]]
| [[19/17]], [[29/26]]
Line 133: Line 142:
|-
|-
| 15
| 15
| 206.897
| 206.9
| [[9/8]]
| [[9/8]]
| [[26/23]]
| [[26/23]]
Line 140: Line 149:
|-
|-
| 16
| 16
| 220.690
| 220.7
| [[25/22]]
| [[25/22]]
| [[17/15]], [[33/29]]
| [[17/15]], [[33/29]]
Line 147: Line 156:
|-
|-
| 17
| 17
| 234.483
| 234.5
| [[8/7]]
| [[8/7]]
| [[31/27]]
| [[31/27]]
Line 154: Line 163:
|-
|-
| 18
| 18
| 248.276
| 248.3
| [[15/13]]
| [[15/13]]
| [[22/19]], [[23/20]], [[38/33]]
| [[22/19]], [[23/20]], [[38/33]]
Line 161: Line 170:
|-
|-
| 19
| 19
| 262.089
| 262.1
| [[7/6]]
| [[7/6]]
| [[29/25]], [[36/31]]
| [[29/25]], [[36/31]]
Line 168: Line 177:
|-
|-
| 20
| 20
| 275.862
| 275.9
| [[75/64]]
| [[75/64]]
| [[20/17]], [[27/23]], [[34/29]]
| [[20/17]], [[27/23]], [[34/29]]
Line 175: Line 184:
|-
|-
| 21
| 21
| 289.655
| 289.7
| [[13/11]], [[32/27]], [[33/28]]
| [[13/11]], [[32/27]], [[33/28]]
|
|
Line 182: Line 191:
|-
|-
| 22
| 22
| 303.448
| 303.4
| [[25/21]]
| [[25/21]]
| [[19/16]], [[31/26]]
| [[19/16]], [[31/26]]
Line 189: Line 198:
|-
|-
| 23
| 23
| 317.241
| 317.2
| [[6/5]]
| [[6/5]]
|
|
Line 196: Line 205:
|-
|-
| 24
| 24
| 331.034
| 331.0
| [[40/33]]
| [[40/33]]
| [[23/19]], [[29/24]]
| [[23/19]], [[29/24]]
Line 203: Line 212:
|-
|-
| 25
| 25
| 344.828
| 344.8
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
Line 210: Line 219:
|-
|-
| 26
| 26
| 358.621
| 358.6
| [[16/13]], [[27/22]]
| [[16/13]], [[27/22]]
| [[38/31]]
| [[38/31]]
Line 217: Line 226:
|-
|-
| 27
| 27
| 372.414
| 372.4
| [[26/21]]
| [[26/21]]
| [[31/25]], [[36/29]]
| [[31/25]], [[36/29]]
Line 224: Line 233:
|-
|-
| 28
| 28
| 386.207
| 386.2
| [[5/4]]
| [[5/4]]
|
|
Line 231: Line 240:
|-
|-
| 29
| 29
| 400.000
| 400.0
| [[44/35]]
| [[44/35]]
| [[24/19]], [[29/23]], [[34/27]]
| [[24/19]], [[29/23]], [[34/27]]
Line 238: Line 247:
|-
|-
| 30
| 30
| 413.793
| 413.8
| [[14/11]], [[33/26]], [[81/64]]
| [[14/11]], [[33/26]], [[81/64]]
| [[19/15]]
| [[19/15]]
Line 245: Line 254:
|-
|-
| 31
| 31
| 427.586
| 427.6
| [[32/25]]
| [[32/25]]
| [[23/18]]
| [[23/18]]
Line 252: Line 261:
|-
|-
| 32
| 32
| 441.379
| 441.4
| [[9/7]], [[35/27]]
| [[9/7]], [[35/27]]
| [[22/17]], [[31/24]], [[40/31]]
| [[22/17]], [[31/24]], [[40/31]]
Line 259: Line 268:
|-
|-
| 33
| 33
| 455.172
| 455.2
| [[13/10]]
| [[13/10]]
| [[30/23]]
| [[30/23]]
Line 266: Line 275:
|-
|-
| 34
| 34
| 468.966
| 469.0
| [[21/16]]
| [[21/16]]
| [[17/13]], [[25/19]], [[38/29]]
| [[17/13]], [[25/19]], [[38/29]]
Line 273: Line 282:
|-
|-
| 35
| 35
| 482.759
| 482.8
| [[33/25]]
| [[33/25]]
|
|
Line 280: Line 289:
|-
|-
| 36
| 36
| 496.552
| 496.6
| [[4/3]]
| [[4/3]]
|
|
Line 287: Line 296:
|-
|-
| 37
| 37
| 510.345
| 510.3
| [[35/26]]
| [[35/26]]
| [[31/23]]
| [[31/23]]
Line 294: Line 303:
|-
|-
| 38
| 38
| 524.138
| 524.1
| [[27/20]]
| [[27/20]]
| [[23/17]]
| [[23/17]]
Line 301: Line 310:
|-
|-
| 39
| 39
| 537.931
| 537.9
| [[15/11]]
| [[15/11]]
| [[26/19]], [[34/25]]
| [[26/19]], [[34/25]]
Line 308: Line 317:
|-
|-
| 40
| 40
| 551.724
| 551.7
| [[11/8]], [[48/35]]
| [[11/8]], [[48/35]]
|
|
Line 315: Line 324:
|-
|-
| 41
| 41
| 565.517
| 565.5
| [[18/13]]
| [[18/13]]
| [[32/23]]
| [[32/23]]
Line 322: Line 331:
|-
|-
| 42
| 42
| 579.310
| 579.3
| [[7/5]]
| [[7/5]]
| [[46/33]]
| [[46/33]]
Line 329: Line 338:
|-
|-
| 43
| 43
| 593.103
| 593.1
| [[45/32]]
| [[45/32]]
| [[24/17]], [[31/22]], [[38/27]]
| [[24/17]], [[31/22]], [[38/27]]
Line 342: Line 351:
| …
| …
|}
|}
== Notation ==
=== Ups and downs notation ===
87edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):
{{Ups and downs sharpness}}
Mapping an arrow to 2\87 rather than 1\87 is an alternative approach which takes advantage of 87edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024.


== Approximation to JI ==
== Approximation to JI ==
Line 348: Line 363:


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3.5
| 2.3.5
| 15625/15552, 67108864/66430125
| 15625/15552, 67108864/66430125
| {{mapping| 87 138 202 }}
| {{Mapping| 87 138 202 }}
| &minus;0.299
| −0.299
| 0.455
| 0.455
| 3.30
| 3.30
Line 359: Line 383:
| 2.3.5.7
| 2.3.5.7
| 245/243, 1029/1024, 3136/3125
| 245/243, 1029/1024, 3136/3125
| {{mapping| 87 138 202 244 }}
| {{Mapping| 87 138 202 244 }}
| +0.070
| +0.070
| 0.752
| 0.752
Line 366: Line 390:
| 2.3.5.7.11
| 2.3.5.7.11
| 245/243, 385/384, 441/440, 3136/3125
| 245/243, 385/384, 441/440, 3136/3125
| {{mapping| 87 138 202 244 301 }}
| {{Mapping| 87 138 202 244 301 }}
| +0.033
| +0.033
| 0.676
| 0.676
Line 373: Line 397:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 196/195, 245/243, 352/351, 364/363, 625/624
| 196/195, 245/243, 352/351, 364/363, 625/624
| {{mapping| 87 138 202 244 301 322 }}
| {{Mapping| 87 138 202 244 301 322 }}
| &minus;0.011
| −0.011
| 0.625
| 0.625
| 4.53
| 4.53
Line 380: Line 404:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374
| {{mapping| 87 138 202 244 301 322 356 }}
| {{Mapping| 87 138 202 244 301 322 356 }}
| &minus;0.198
| −0.198
| 0.738
| 0.738
| 5.35
| 5.35
Line 387: Line 411:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363
| {{mapping| 87 138 202 244 301 322 356 370 }}
| {{Mapping| 87 138 202 244 301 322 356 370 }}
| &minus;0.348
| −0.348
| 0.796
| 0.796
| 5.77
| 5.77
{{comma basis end}}
|}


=== 13-limit detempering ===
=== 13-limit detempering ===
Line 397: Line 421:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
Line 460: Line 491:
|-
|-
| 3
| 3
| 18\87<br />(11\87)
| 18\87<br>(11\87)
| 248.276<br />(151.724)
| 248.276<br>(151.724)
| 15/13<br />(12/11)
| 15/13<br>(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
| 3
| 3
| 23\87<br />(6\87)
| 23\87<br>(6\87)
| 317.241<br />(82.759)
| 317.241<br>(82.759)
| 6/5<br />(21/20)
| 6/5<br>(21/20)
| [[Tritikleismic]]
| [[Tritikleismic]]
|-
|-
| 3
| 3
| 28\87<br />(1\87)
| 28\87<br>(1\87)
| 386.207<br />(13.793)
| 386.207<br>(13.793)
| 5/4<br />(126/125)
| 5/4<br>(126/125)
| [[Mutt]]
| [[Mutt]]
|-
|-
| 3
| 3
| 36\87<br />(7\87)
| 36\87<br>(7\87)
| 496.552<br />(96.552)
| 496.552<br>(96.552)
| 4/3<br />(18/17~19/18)
| 4/3<br>(18/17~19/18)
| [[Misty]]
| [[Misty]]
|-
|-
| 29
| 29
| 28\87<br />(1\87)
| 28\87<br>(1\87)
| 386.207<br />(13.793)
| 386.207<br>(13.793)
| 5/4<br />(121/120)
| 5/4<br>(121/120)
| [[Mystery]]
| [[Mystery]]
{{rank-2 end}}
|}
{{orf}}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


87 can serve as a MOS in these:
87 can serve as a mos in these:


* [[Avicenna (temperament)|Avicenna]] ([[Breed|87&amp;270]]) {{multival| 24 -9 -66 12 27 … }}
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]])  
* [[Breed|87&amp;494]] {{multival| 51 -1 -133 11 32 … }}
* [[Breed|87 & 494]]  


== Scales ==
== Scales ==
=== MOS scales ===
=== Harmonic scales ===
{{main|List of MOS scales in 87edo}}
87edo very accurately approximates [[Overtone scale|modes]] up to 8 of the [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.
 
==== (Mode 6) ====
{| class="wikitable center-all"
|-
! Overtones
| 12
| 14
| 16
| 18
| 20
| 22
| 24
|-
! JI Ratios
| 1/1
| 7/6
| 4/3
| 3/2
| 5/3
| 11/6
| 2/1
|-
! … in cents
| 0.0
| 266.9
| 498.0
| 702.0
| 884.4
| 1049.4
| 1200.0
|-
! Degrees in 87edo
| 0
| 19
| 36
| 51
| 64
| 76
| 87
|-
! … in cents
| 0.0
| 262.1
| 496.6
| 703.4
| 882.8
| 1048.3
| 1200.0
|}


=== Harmonic scale ===
The scale in adjacent steps is '''19 17 15 13 12 11'''.  
87edo accurately approximates the mode 8 of [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently.  


==== Mode 8 ====
==== (Mode 8) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 562: Line 641:
|}
|}


* The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8.  
The scale in adjacent steps is '''15 13 12 11 10 9 9 8'''.  


==== Mode 12 ====
==== (Mode 12) ====
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 643: Line 722:
|}
|}


* The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5.  
The scale in adjacent steps is '''10 9 9 8 7 7 6 6 6 6 5'''.  
* 13, 15, 16, 18, 20, and 22 are close matches.  
 
* 14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
13, 15, 16, 18, 20, and 22 are close matches.  
 
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered.
 
=== Mos scales ===
{{main|List of MOS scales in 87edo}}
* [[Kleismic]][7]: 5 18 5 18 5 18 18
* Kleismic[15]: 5 8 5 5 5 8 5 5 5 8 5 5 5 8 5
* Kleismic[19]: 5 5 3 5 5 5 5 3 5 5 5 3 5 5 5 5 3 5 5
* Kleismic[34]: 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3
* [[Misty]][6]: 22 7 22 7 22 7
* Misty[12]: 7 8 7 7 7 8 7 7 7 8 7 7
* Misty[15]: 7 7 1 7 7 7 7 1 7 7 7 7 1 7 7
* Misty[27]: 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1 1 6 1 6 1 6 1 6 1
* Misty[39]: 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 1 5 1 1 1 5 1 1 5 1
* [[Luna]][6]: 17 14 14 14 14 14
* Luna[13]: 3 11 3 11 3 11 3 11 3 11 3 11 3
* Luna[19]: 3 8 3 3 8 3 3 8 3 3 3 8 3 3 8 3 3 8 3
* Luna[25]: 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3 3 5 3 3
* Luna[31]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
* Luna[56]: 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2
* [[Rodan]][5]: 17 17 17 17 19
* Rodan[11]: 2 15 2 15 2 15 2 15 2 15 2
* Rodan[16]: 2 13 2 2 13 2 2 2 13 2 2 13 2 2 13 2
* Rodan[21]: 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2 2 11 2 2
* Rodan[36]: 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2 2 2 2 5 2 2 2
* Rodan[41]: 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2
 
=== Other scales ===
* [[Sequar5m]]
 
== Instruments ==
* [[Lumatone mapping for 87edo]]
* [[Skip fretting system 87 2 17]]


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Stomach Book}}
* [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent''] (2024) – microtonal cover in 87edo by [[Bryan Deister]] (2025)
=== 21st century ===
; [[ALLY195]]
* [https://www.bilibili.com/video/BV16h411g7QM/ ''Root note and subharmonic series cadence – 103EDO, 87EDO, 94EDO''] (2023)
* [https://www.bilibili.com/video/BV1N84y1T792/ ''A comparison between 87edo and 12edo''] (2023)
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025)
* [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025)
* [https://www.youtube.com/shorts/8mHBYBfRjy4 ''87edo improv''] (2026-07-11)
* [https://www.youtube.com/shorts/wHPyCHQn55A ''87edo improv''] (2026-09-28)
; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Pianodactyl'' (archived 2010) &ndash; [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] &ndash; rodan[26] in 87edo tuning
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning


[[Category:Listen]]
[[Category:Clyde]]
[[Category:Clyde]]
[[Category:Countercata]]
[[Category:Countercata]]
[[Category:Hemithirds]]
[[Category:Hemithirds]]
[[Category:Listen]]
[[Category:Mystery]]
[[Category:Mystery]]
[[Category:Rodan]]
[[Category:Rodan]]
[[Category:Tritikleismic]]
[[Category:Tritikleismic]]
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